He called on current leader, Anas Sarwar, to drop his opposition and make a positive case for constitutional reform.
Mr McLeish said: “Merely saying no to a second independence referendum has never been a vote winner. It sounds negative, is interpreted as a denial of democracy and smacks of political panic.
“This has been a problem for Labour, especially in the absence of a well thought-out alternative.
“Now is the time for the party to be more self-confident. A new referendum could be some years away.
“Labour should drop its blanket opposition to another referendum. It will happen some time.
“The party must focus on other questions to be asked of the electorate and to ensure that other alternatives, not just independence, are part of any future debate and public vote.
“The real question is whether the Union, the Tory Government and the Westminster Parliament can produce the radical reforms needed to accommodate an ambitious nation like Scotland in a reasonable timescale.”
SNP depute leader, Keith Brown, said Mr Sarwar should listen to the former first minister and it was time for Scottish Labour to “finally accept” Scots have the right to determine their own future.
He said: “Anas Sarwar should listen to Henry McLeish, and offer the Labour Party's support for a post-pandemic independence referendum - instead of backing Boris Johnson's doomed attempts to deny democracy.
"The Labour Party ruined its reputation in Scotland by siding with the Tories on austerity cuts, and in the Better Together campaign which Anas Sarwar ran for Labour, and now by backing Brexit.
“Instead of working hand-in-hand with Boris Johnson, yet again, they must finally accept people in Scotland have a right to determine their own future.”
On the SNP, Mr McLeish said that the recent spat between Alex Salmond and Nicola Sturgeon had “undermined” Scottish politics and warned against the planned “legitimate gaming of the system” by the Alba party who are hoping to take advantage of the regional list system to return more pro-independence MSPs.
He said: “Fourteen years in government has taken its toll on the SNP, which appears less assured and riven with internal divisions.
“The SNP and Alba will now compete for independence votes in the aftermath of a vicious public, political and personal spat between the First Minister and her predecessor.
“Our politics has been undermined. Weaknesses have been exposed in our parliament and government. Critics of devolution at Westminster have been strengthened.
“Despite this, the SNP is likely to remain the largest party, with Alex Salmond’s new party contributing seats in support of independence.
“It would be foolish to underestimate the political significance of this legitimate gaming of the system. It is time to see Scotland’s future through the prism of a longer timescale.”